Distinguish between a pole and an essential singularity?

If the Laurent series has only finitely many terms with negative powers of z - c, then the singularity is a pole. The biggest negative exponent is the order of the pole.

Example: Singularities of rational functions with no common factors in its numerator and denominator. (These come from setting the denominator equal to 0.)

If the Laurent series has infinitely many terms with negative powers of z - c, then the singularity is essential.

Example: e^(1/z) = 1 + (1/z) + (1/2!) 1/z^2 + ... has an essential singularity at z = 0.